I dont want to discourage you from learning math, but I think this isnt a good place spend too much thought at first.
I've heard a bunch of people say that category theory is almost like a 'bird's eye view of mathematics'. That many of its concepts unify many mathematical concepts under one umbrella, and reveal the similarities and connections between mathematical subdisciplines.
The main reason this interests me is that I'd like to learn math much differently than its taught.
To me, to learn the same concept under different subdisciplines with different names, is O(N) learning.
However, if someone put together a category theory book where they list a concept, and then they do this:
"Now, any mathematician can (easily) see that every major area of mathematics is a category.
- In Set Theory the arrows are functions
- in Topology they are continuous functions
- in Group Theory homomorphisms
- in Linear Algebra they are linear transformations
- in Differentiable Geometry they’re smooth maps, and so on…
But what’s important is that we shifted from focusing on the objects to focusing on the functions, the ways in which we transform the objects. This is basically the category-theoretical perspective: it’s the functions that matter."
(Taken from this blogpost: https://catsinthejungle.wordpress.com/2008/11/15/category-th... )
Do I understand any of that? No. But what attracts me to this format is it (MAYBE...) approximates O(1) learning in math, so long as it can also point out, for every concept in every field, how they differ/relate to every other concept.
So what this does is it helps teach me the 'unifiers' of mathematics, and helps me see connections between fields which can be useful for creativity. E.g. I realize I'm doing X algorithm here from linear algebra and that is O(N), whereas if I just shift fields, and use Y algorithm from Group theory I might make it O(log n) because it differs in this way or whatever it is.
Besides that I see it as a potentially efficient way of learning a lot of math.
While a shortcut to understanding sounds nice, I don't think you'll find it by focusing on abstract formalism.
In this case, seeing the connections across the disciplines isn't necessarily something I see or am pursuing because I'm searching for a shortcut (although I did mention O(1)), I just see that as a side benefit. What appeals to me more is developing the intuition you mention via the analogies. I feel like if the idea is restated in different ways (not necessarily just terms, but perhaps visualizations/re-conceptualizations) a more holistic/intuitive/fuzzy idea emerges rather than anything rigorous necessarily, but you know a lot more math than me so maybe I'm saying nonsense.